Optimal. Leaf size=26 \[ \frac {\csc (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2606} \[ \frac {\csc (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2606
Rubi steps
\begin {align*} \int \cot ^3(a+b x) \csc (a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (a+b x)\right )}{b}\\ &=\frac {\csc (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.00 \[ \frac {\csc (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 38, normalized size = 1.46 \[ \frac {3 \, \cos \left (b x + a\right )^{2} - 2}{3 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 25, normalized size = 0.96 \[ \frac {3 \, \sin \left (b x + a\right )^{2} - 1}{3 \, b \sin \left (b x + a\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 60, normalized size = 2.31 \[ \frac {-\frac {\cos ^{4}\left (b x +a \right )}{3 \sin \left (b x +a \right )^{3}}+\frac {\cos ^{4}\left (b x +a \right )}{3 \sin \left (b x +a \right )}+\frac {\left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 25, normalized size = 0.96 \[ \frac {3 \, \sin \left (b x + a\right )^{2} - 1}{3 \, b \sin \left (b x + a\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 22, normalized size = 0.85 \[ \frac {{\sin \left (a+b\,x\right )}^2-\frac {1}{3}}{b\,{\sin \left (a+b\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.26, size = 42, normalized size = 1.62 \[ \begin {cases} \frac {2}{3 b \sin {\left (a + b x \right )}} - \frac {\cos ^{2}{\left (a + b x \right )}}{3 b \sin ^{3}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{3}{\relax (a )}}{\sin ^{4}{\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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